LIRA (Line Resonance Analysis System) is based on transmission line theory, an established and well documented theory that is at the base of two other existing cable fail detection techniques known as “Time Domain Reflectometry” (TDR) and “Frequency Domain Reflectometry” (FDR). Differences and limitations of these two existing techniques are explained in the following.
A transmission line is the part of an electrical circuit providing a link between a generator and a load. The behavior of a transmission line depends by its length in comparison with the wavelength λ of the electrical signal traveling into it. The wavelength is defined as:λ=v/f  (1)where v is the speed of the electric signal in the wire (also called the phase velocity) and f the frequency of the signal.
When the transmission line length is much lower than the wavelength, as it happens when the cable is short (i.e. few meters) and the signal frequency is low (i.e. few KHz), the line has no influence on the circuit behavior. Then the circuit impedance (Zin), as seen from the generator side, is equal to the load impedance at any time.
However, if the line length is higher than the signal wavelength, (L≧λ), the line characteristics take an important role and the circuit impedance seen from the generator does not match the load, except for some very particular cases.
The voltage V and the current I along the cable are governed by the following differential equations, known as the telephonists equations:
                                                        ⅆ              2                        ⁢            V                                ⅆ                          z              2                                      =                              (                          R              +                              jω                ⁢                                                                  ⁢                L                                      )                    ⁢                      (                          G              +                              jω                ⁢                                                                  ⁢                C                                      )                    ⁢          V                                    (        2        )                                                                    ⅆ              2                        ⁢            I                                ⅆ                          z              2                                      =                              (                          R              +                              jω                ⁢                                                                  ⁢                L                                      )                    ⁢                      (                          G              +                              jω                ⁢                                                                  ⁢                C                                      )                    ⁢          I                                    (        3        )            where ω is the signal radial frequency, R is the conductor resistance, L is the inductance, C the capacitance and G the insulation conductivity, all relative to a unit of cable length.
These four parameters completely characterize the behavior of a cable when a high frequency signal is passing through it. In transmission line theory, the line behavior is normally studied as a function of two complex parameters. The first is the propagation functionγ=√{square root over ((R+jωL)(G+jωC))}{square root over ((R+jωL)(G+jωC))}  (4)often written asγ=αjβ  (5)where the real part α is the line attenuation constant and the imaginary part β is the propagation constant, which is also related to the phase velocity v, radial frequency ω and wavelength λ through:
                                                        β              =                            ⁢                                                2                  ⁢                  π                                λ                                                                                        =                            ⁢                              ω                v                                                                        (        6        )            
The second parameter is the characteristic impedance
                              Z          0                =                                            R              +                              jω                ⁢                                                                  ⁢                L                                                    G              +                              jω                ⁢                                                                  ⁢                C                                                                        (        7        )            
Using (4) and (7) and solving the differential equations (2) and (3), the line impedance for a cable at distance d from the end is:
                                                                        Z                d                            =                            ⁢                                                V                  ⁡                                      (                    d                    )                                                                    I                  ⁡                                      (                    d                    )                                                                                                                          =                            ⁢                                                Z                  0                                ⁢                                                      1                    +                                          Γ                      d                                                                            1                    -                                          Γ                      d                                                                                                                              (        8        )            Where Γd is the Generalized Reflection CoefficientΓd=ΓLe−2γd  (9)and ΓL is the Load Reflection Coefficient
                              Γ          L                =                                            Z              L                        -                          Z              0                                                          Z              L                        +                          Z              0                                                          (        10        )            
In (10) ZL is the impedance of the load connected at the cable end.
From eqs. (8), (9) and (10), it is easy to see that when the load matches the characteristic impedance, ΓL=Γd=0 and then Zd=Z0=ZL for any length and frequency. In all the other cases, the line impedance is a complex variable governed by eq. (8), which has the shape of the curves in FIG. 1 (amplitude and phase as a function of frequency).
Existing methods based on transmission line theory try to localize local cable failures (no global degradation assessment is possible) by a measure of V (equation (2)) as a function of time and evaluating the time delay from the incident wave to the reflected wave. Examples of such methods are found in U.S. Pat. Nos. 4,307,267 and 4,630,228, and in US applications 2004/0039976 and 2005/0057259. Line attenuation and environment noise in real environments limit the sensitivity of such methods, preventing the possibility to detect degradations at an early stage, especially for cables longer than a few kilometers. In addition, no global cable condition assessment is possible, which is important for cable residual life estimation in harsh environment applications (for example nuclear and aerospace applications).